Civil Engineering Reference
In-Depth Information
Figure 4.28
Mass modelling of RC frame in Figure 4.2: 3D structure (
left
), FE model without (
middle
) and with
(
right
) masses
Key
: Cubic markers indicate the location of lumped masses
metric discretization adopted. For multi-storey framed buildings, it is commonly assumed that structural
masses are lumped at storey levels. Figure 4.28 shows the lumped mass modelling of the SPEAR frame
fi rst presented in Figure 4.2. The masses are applied at beam-to-column connections at all storeys. It
is instructive to note that the actual fundamental period of the full-scale RC frame is higher than the
values computed by eigenvalue analysis of the refi ned 3D-FE model shown in Figures 4.22 and 4.28 .
Variations between analytical and experimental periods may be caused by micro- cracks, which are
inevitable in RC members. However, the smallest differences, on average less than 4% (Jeong and
Elnashai, 2005), are found for the FE models that include rigid diaphragms and shear joint models
presented in Sections 4.5.3.3 and 4.5.3.1 , respectively.
In commercial software packages for earthquake analysis, the mass of each beam is automatically
lumped at the member end nodes. Further, in FE modelling of framed structures with ' diaphragm con-
straints' and master nodes, which have been presented in Section 4.5.3.3, two translational masses and
one rotational mass about the vertical axis are necessary for 3D dynamic analysis. The two translational
masses are applied along the principal directions of the framed systems. For two-dimensional frame
models, only one translational mass is required.
Bridge piers are modelled as 2D or 3D fl exural systems. Most of the mass is associated with the
deck. Bridge decks are often represented as linear elastic systems. The mass of the majority of bridge
piers is a small proportion of the upper deck and can therefore be neglected or lumped with the deck
mass. If the stresses in the pier are critically affected by its own vibration modes, or its mass is non-
negligible, its mass is represented separate from the deck (Figure 4.29). When decks are suffi ciently
wide, their torsional modes may be important and an adequate mass representation along the width of
the deck is necessary. Accurate representations of rotational inertia mass should be included particularly
for large decks in single-pier bridges to account for higher mode effects that may be introduced by
near - fi eld earthquake strong motions.
Whenever vertical effects of earthquake ground motion (Sections 3.4.6 and 3.4.7) are of interest, the
vertical translational mass has to be included in the FE model. It may also be necessary to model masses
on the beams to capture the effects of their vibrations under vertical motion.
The description above has focused on the representation of structural masses for dynamic analyses.
Non-structural masses may also be present in bridges and buildings, e.g. in the form of attached machin-
ery, water tanks or other heavy electrical and mechanical equipment. These should be included in the
model.
